tm: comparison thys/UF

equal deleted inserted replaced

-:6e7244ae43b8
+:745547bdc1c7
 theory UF_Rec
 imports Recs Turing_Hoare
 begin
-section {* Universal Function *}
-text{* coding of the configuration *}
+section {* Universal Function in HOL *}
 text {*
 @{text "Entry sr i"} returns the @{text "i"}-th entry of a list of natural
 numbers encoded by number @{text "sr"} using Godel's coding.
 *}
 fun Entry where
 "Entry sr i = Lo sr (Pi (Suc i))"
-definition
-"rec_entry = CN rec_lo [Id 2 0, CN rec_pi [CN S [Id 2 1]]]"
-lemma entry_lemma [simp]:
-"rec_eval rec_entry [sr, i] = Entry sr i"
-by(simp add: rec_entry_def)
 fun Listsum2 :: "nat list \<Rightarrow> nat \<Rightarrow> nat"
 where
 "Listsum2 xs 0 = 0"
 | "Listsum2 xs (Suc n) = Listsum2 xs n + xs ! n"
-fun rec_listsum2 :: "nat \<Rightarrow> nat \<Rightarrow> recf"
-where
-"rec_listsum2 vl 0 = CN Z [Id vl 0]"
-| "rec_listsum2 vl (Suc n) = CN rec_add [rec_listsum2 vl n, Id vl n]"
-lemma listsum2_lemma [simp]:
-"length xs = vl \<Longrightarrow> rec_eval (rec_listsum2 vl n) xs = Listsum2 xs n"
-by (induct n) (simp_all)
 text {*
 @{text "Strt"} corresponds to the @{text "strt"} function on page 90 of the
-B book, but this definition generalises the original one to deal with multiple
+B book, but this definition generalises the original one in order to deal
-input arguments.
+with multiple input arguments. *}
-*}
 fun Strt' :: "nat list \<Rightarrow> nat \<Rightarrow> nat"
 where
 "Strt' xs 0 = 0"
 | "Strt' xs (Suc n) = (let dbound = Listsum2 xs n + n
 in Strt' xs n + (2 ^ (xs ! n + dbound) - 2 ^ dbound))"
 fun Strt :: "nat list \<Rightarrow> nat"
 where
 "Strt xs = (let ys = map Suc xs in Strt' ys (length ys))"
+text {* The @{text "Scan"} function on page 90 of B book. *}
+fun Scan :: "nat \<Rightarrow> nat"
+where
+"Scan r = r mod 2"
+text {* The @{text Newleft} and @{text Newright} functions on page 91 of B book. *}
+fun Newleft :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+"Newleft p r a = (if a = 0 \<or> a = 1 then p
+else if a = 2 then p div 2
+else if a = 3 then 2 * p + r mod 2
+else p)"
+fun Newright :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+"Newright p r a  = (if a = 0 then r - Scan r
+else if a = 1 then r + 1 - Scan r
+else if a = 2 then 2 * r + p mod 2
+else if a = 3 then r div 2
+else r)"
+text {*
+The @{text "Actn"} function given on page 92 of B book, which is used to
+fetch Turing Machine intructions. In @{text "Actn m q r"}, @{text "m"} is
+the Goedel coding of a Turing Machine, @{text "q"} is the current state of
+Turing Machine, @{text "r"} is the right number of Turing Machine tape. *}
+fun Actn :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+"Actn m q r = (if q \<noteq> 0 then Entry m (4 * (q - 1) + 2 * Scan r) else 4)"
+fun Newstat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+"Newstat m q r = (if q \<noteq> 0 then Entry m (4 * (q - 1) + 2 * Scan r + 1) else 0)"
+fun Trpl :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+"Trpl p q r = (Pi 0) ^ p * (Pi 1) ^ q * (Pi 2) ^ r"
+fun Left where
+"Left c = Lo c (Pi 0)"
+fun Right where
+"Right c = Lo c (Pi 2)"
+fun Stat where
+"Stat c = Lo c (Pi 1)"
+lemma mod_dvd_simp:
+"(x mod y = (0::nat)) = (y dvd x)"
+by(auto simp add: dvd_def)
+lemma Trpl_Left [simp]:
+"Left (Trpl p q r) = p"
+apply(simp)
+apply(subst Lo_def)
+apply(subst dvd_eq_mod_eq_0[symmetric])
+apply(simp)
+apply(auto)
+thm Lo_def
+thm mod_dvd_simp
+apply(simp add: left.simps trpl.simps lo.simps
+loR.simps mod_dvd_simp, auto simp: conf_decode1)
+apply(case_tac "Pi 0 ^ l * Pi (Suc 0) ^ st * Pi (Suc (Suc 0)) ^ r",
+auto)
+apply(erule_tac x = l in allE, auto)
+fun Inpt :: "nat \<Rightarrow> nat list \<Rightarrow> nat"
+where
+"Inpt m xs = Trpl 0 1 (Strt xs)"
+fun Newconf :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+"Newconf m c = Trpl (Newleft (Left c) (Right c) (Actn m (Stat c) (Right c)))
+(Newstat m (Stat c) (Right c))
+(Newright (Left c) (Right c) (Actn m (Stat c) (Right c)))"
+text {*
+@{text "Conf k m r"} computes the TM configuration after @{text "k"} steps of execution
+of TM coded as @{text "m"} starting from the initial configuration where the left
+number equals @{text "0"}, right number equals @{text "r"}. *}
+fun Conf :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+"Conf 0 m r  = Trpl 0 1 r"
+| "Conf (Suc k) m r = Newconf m (Conf k m r)"
+text {*
+@{text "Nstd c"} returns true if the configuration coded
+by @{text "c"} is not a stardard final configuration. *}
+fun Nstd :: "nat \<Rightarrow> bool"
+where
+"Nstd c = (Stat c \<noteq> 0 \<or>
+Left c \<noteq> 0 \<or>
+Right c \<noteq> 2 ^ (Lg (Suc (Right c)) 2) - 1 \<or>
+Right c = 0)"
+text{*
+@{text "Nostop t m r"} means that afer @{text "t"} steps of
+execution the TM coded by @{text "m"} is not at a stardard
+final configuration. *}
+fun Nostop :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"
+where
+"Nostop t m r = Nstd (Conf t m r)"
+fun Value where
+"Value x = (Lg (Suc x) 2) - 1"
+text{*
+@{text "rec_halt"} is the recursive function calculating the steps a TM needs to execute before
+to reach a stardard final configuration. This recursive function is the only one
+using @{text "Mn"} combinator. So it is the only non-primitive recursive function
+needs to be used in the construction of the universal function @{text "rec_uf"}. *}
+fun Halt :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+"Halt m r = (LEAST t. \<not> Nostop t m r)"
+fun UF :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+"UF c m = Value (Right (Conf (Halt c m) c m))"
+section {* Coding of Turing Machines *}
+text {*
+The purpose of this section is to construct the coding function of Turing
+Machine, which is going to be named @{text "code"}. *}
+fun bl2nat :: "cell list \<Rightarrow> nat \<Rightarrow> nat"
+where
+"bl2nat [] n = 0"
+| "bl2nat (Bk # bl) n = bl2nat bl (Suc n)"
+| "bl2nat (Oc # bl) n = 2 ^ n + bl2nat bl (Suc n)"
+fun bl2wc :: "cell list \<Rightarrow> nat"
+where
+"bl2wc xs = bl2nat xs 0"
+lemma bl2nat_double [simp]:
+"bl2nat xs (Suc n) = 2 * bl2nat xs n"
+apply(induct xs arbitrary: n)
+apply(auto)
+apply(case_tac a)
+apply(auto)
+done
+lemma bl2nat_simps1 [simp]:
+shows "bl2nat (Bk \<up> y) n = 0"
+by (induct y) (auto)
+lemma bl2nat_simps2 [simp]:
+shows "bl2nat (Oc \<up> y) 0 = 2 ^ y - 1"
+apply(induct y)
+apply(auto)
+apply(case_tac "(2::nat)^ y")
+apply(auto)
+done
+fun Trpl_code :: "config \<Rightarrow> nat"
+where
+"Trpl_code (st, l, r) = Trpl (bl2wc l) st (bl2wc r)"
+fun action_map :: "action \<Rightarrow> nat"
+where
+"action_map W0 = 0"
+| "action_map W1 = 1"
+| "action_map L = 2"
+| "action_map R = 3"
+| "action_map Nop = 4"
+fun action_map_iff :: "nat \<Rightarrow> action"
+where
+"action_map_iff (0::nat) = W0"
+| "action_map_iff (Suc 0) = W1"
+| "action_map_iff (Suc (Suc 0)) = L"
+| "action_map_iff (Suc (Suc (Suc 0))) = R"
+| "action_map_iff n = Nop"
+fun block_map :: "cell \<Rightarrow> nat"
+where
+"block_map Bk = 0"
+| "block_map Oc = 1"
+fun Goedel_code' :: "nat list \<Rightarrow> nat \<Rightarrow> nat"
+where
+"Goedel_code' [] n = 1"
+| "Goedel_code' (x # xs) n = (Pi n) ^ x * Goedel_code' xs (Suc n) "
+fun Goedel_code :: "nat list \<Rightarrow> nat"
+where
+"Goedel_code xs = 2 ^ (length xs) * (Goedel_code' xs 1)"
+fun modify_tprog :: "instr list \<Rightarrow> nat list"
+where
+"modify_tprog [] =  []"
+| "modify_tprog ((a, s) # nl) = action_map a # s # modify_tprog nl"
+text {* @{text "Code tp"} gives the Goedel coding of TM program @{text "tp"}. *}
+fun Code :: "instr list \<Rightarrow> nat"
+where
+"Code tp = Goedel_code (modify_tprog tp)"
+section {* Universal Function as Recursive Functions *}
+definition
+"rec_entry = CN rec_lo [Id 2 0, CN rec_pi [CN S [Id 2 1]]]"
+fun rec_listsum2 :: "nat \<Rightarrow> nat \<Rightarrow> recf"
+where
+"rec_listsum2 vl 0 = CN Z [Id vl 0]"
+| "rec_listsum2 vl (Suc n) = CN rec_add [rec_listsum2 vl n, Id vl n]"
 fun rec_strt' :: "nat \<Rightarrow> nat \<Rightarrow> recf"
 where
 "rec_strt' xs 0 = Z"
 | "rec_strt' xs (Suc n) =
 fun rec_strt :: "nat \<Rightarrow> recf"
 where
 "rec_strt xs = CN (rec_strt' xs xs) (rec_map S xs)"
-lemma strt'_lemma [simp]:
-"length xs = vl \<Longrightarrow> rec_eval (rec_strt' vl n) xs = Strt' xs n"
-by (induct n) (simp_all add: Let_def)
-lemma map_suc:
-"map (\<lambda>x. Suc (xs ! x)) [0..<length xs] = map Suc xs"
-proof -
-have "Suc \<circ> (\<lambda>x. xs ! x) = (\<lambda>x. Suc (xs ! x))" by (simp add: comp_def)
-then have "map (\<lambda>x. Suc (xs ! x)) [0..<length xs] = map (Suc \<circ> (\<lambda>x. xs ! x)) [0..<length xs]" by simp
-also have "... = map Suc (map (\<lambda>x. xs ! x) [0..<length xs])" by simp
-also have "... = map Suc xs" by (simp add: map_nth)
-finally show "map (\<lambda>x. Suc (xs ! x)) [0..<length xs] = map Suc xs" .
-qed
-lemma strt_lemma [simp]:
-"length xs = vl \<Longrightarrow> rec_eval (rec_strt vl) xs = Strt xs"
-by (simp add: comp_def map_suc[symmetric])
-text {* The @{text "Scan"} function on page 90 of B book. *}
-fun Scan :: "nat \<Rightarrow> nat"
-where
-"Scan r = r mod 2"
 definition
 "rec_scan = CN rec_mod [Id 1 0, constn 2]"
-lemma scan_lemma [simp]:
-"rec_eval rec_scan [r] = r mod 2"
-by(simp add: rec_scan_def)
-text {* The @{text Newleft} and @{text Newright} functions on page 91 of B book. *}
-fun Newleft :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-"Newleft p r a = (if a = 0 \<or> a = 1 then p
-else if a = 2 then p div 2
-else if a = 3 then 2 * p + r mod 2
-else p)"
-fun Newright :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-"Newright p r a  = (if a = 0 then r - Scan r
-else if a = 1 then r + 1 - Scan r
-else if a = 2 then 2 * r + p mod 2
-else if a = 3 then r div 2
-else r)"
 definition
 "rec_newleft =
 (let cond1 = CN rec_disj [CN rec_eq [Id 3 2, Z], CN rec_eq [Id 3 2, constn 1]] in
 let cond2 = CN rec_eq [Id 3 2, constn 2] in
 CN rec_if [condn 0, case0,
 CN rec_if [condn 1, case1,
 CN rec_if [condn 2, case2,
 CN rec_if [condn 3, case3, Id 3 1]]]])"
-lemma newleft_lemma [simp]:
-"rec_eval rec_newleft [p, r, a] = Newleft p r a"
-by (simp add: rec_newleft_def Let_def)
-lemma newright_lemma [simp]:
-"rec_eval rec_newright [p, r, a] = Newright p r a"
-by (simp add: rec_newright_def Let_def)
-text {*
-The @{text "Actn"} function given on page 92 of B book, which is used to
-fetch Turing Machine intructions. In @{text "Actn m q r"}, @{text "m"} is
-the Goedel coding of a Turing Machine, @{text "q"} is the current state of
-Turing Machine, @{text "r"} is the right number of Turing Machine tape.
-*}
-fun Actn :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-"Actn m q r = (if q \<noteq> 0 then Entry m (4 * (q - 1) + 2 * Scan r) else 4)"
 definition
 "rec_actn = (let add1 = CN rec_mult [constn 4, CN rec_pred [Id 3 1]] in
 let add2 = CN rec_mult [constn 2, CN rec_scan [Id 3 2]] in
 let entry = CN rec_entry [Id 3 0, CN rec_add [add1, add2]]
 in CN rec_if [Id 3 1, entry, constn 4])"
-lemma actn_lemma [simp]:
-"rec_eval rec_actn [m, q, r] = Actn m q r"
-by (simp add: rec_actn_def)
-fun Newstat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-"Newstat m q r = (if q \<noteq> 0 then Entry m (4 * (q - 1) + 2 * Scan r + 1) else 0)"
 definition rec_newstat :: "recf"
 where
 "rec_newstat = (let add1 = CN rec_mult [constn 4, CN rec_pred [Id 3 1]] in
 let add2 = CN S [CN rec_mult [constn 2, CN rec_scan [Id 3 2]]] in
 let entry = CN rec_entry [Id 3 0, CN rec_add [add1, add2]]
 in CN rec_if [Id 3 1, entry, Z])"
-lemma newstat_lemma [simp]:
-"rec_eval rec_newstat [m, q, r] = Newstat m q r"
-by (simp add: rec_newstat_def)
-fun Trpl :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-"Trpl p q r = (Pi 0) ^ p * (Pi 1) ^ q * (Pi 2) ^ r"
 definition
 "rec_trpl = CN rec_mult [CN rec_mult
 [CN rec_power [constn (Pi 0), Id 3 0],
 CN rec_power [constn (Pi 1), Id 3 1]],
 CN rec_power [constn (Pi 2), Id 3 2]]"
-lemma trpl_lemma [simp]:
-"rec_eval rec_trpl [p, q, r] = Trpl p q r"
-by (simp add: rec_trpl_def)
-fun Left where
-"Left c = Lo c (Pi 0)"
 definition
 "rec_left = CN rec_lo [Id 1 0, constn (Pi 0)]"
-lemma left_lemma [simp]:
-"rec_eval rec_left [c] = Left c"
-by(simp add: rec_left_def)
-fun Right where
-"Right c = Lo c (Pi 2)"
 definition
 "rec_right = CN rec_lo [Id 1 0, constn (Pi 2)]"
-lemma right_lemma [simp]:
-"rec_eval rec_right [c] = Right c"
-by(simp add: rec_right_def)
-fun Stat where
-"Stat c = Lo c (Pi 1)"
 definition
 "rec_stat = CN rec_lo [Id 1 0, constn (Pi 1)]"
-lemma stat_lemma [simp]:
-"rec_eval rec_stat [c] = Stat c"
-by(simp add: rec_stat_def)
-fun Inpt :: "nat \<Rightarrow> nat list \<Rightarrow> nat"
-where
-"Inpt m xs = Trpl 0 1 (Strt xs)"
-fun Newconf :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-"Newconf m c = Trpl (Newleft (Left c) (Right c) (Actn m (Stat c) (Right c)))
-(Newstat m (Stat c) (Right c))
-(Newright (Left c) (Right c) (Actn m (Stat c) (Right c)))"
 definition
 "rec_newconf = (let act = CN rec_actn [Id 2 0, CN rec_stat [Id 2 1], CN rec_right [Id 2 1]] in
 let left = CN rec_left [Id 2 1] in
 let right = CN rec_right [Id 2 1] in
 let one = CN rec_newleft [left, right, act] in
 let two = CN rec_newstat [Id 2 0, stat, right] in
 let three = CN rec_newright [left, right, act]
 in CN rec_trpl [one, two, three])"
-lemma newconf_lemma [simp]:
-"rec_eval rec_newconf [m, c] = Newconf m c"
-by (simp add: rec_newconf_def Let_def)
-text {*
-@{text "Conf k m r"} computes the TM configuration after @{text "k"} steps of execution
-of TM coded as @{text "m"} starting from the initial configuration where the left
-number equals @{text "0"}, right number equals @{text "r"}.
-*}
-fun Conf :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-"Conf 0 m r  = Trpl 0 1 r"
-| "Conf (Suc k) m r = Newconf m (Conf k m r)"
 definition
 "rec_conf = PR (CN rec_trpl [constn 0, constn 1, Id 2 1])
 (CN rec_newconf [Id 4 2 , Id 4 1])"
-lemma conf_lemma [simp]:
-"rec_eval rec_conf [k, m, r] = Conf k m r"
-by(induct k) (simp_all add: rec_conf_def)
-text {*
-@{text "Nstd c"} returns true if the configuration coded
-by @{text "c"} is not a stardard final configuration.
-*}
-fun Nstd :: "nat \<Rightarrow> bool"
-where
-"Nstd c = (Stat c \<noteq> 0 \<or>
-Left c \<noteq> 0 \<or>
-Right c \<noteq> 2 ^ (Lg (Suc (Right c)) 2) - 1 \<or>
-Right c = 0)"
 definition
 "rec_nstd = (let disj1 = CN rec_noteq [rec_stat, constn 0] in
 let disj2 = CN rec_noteq [rec_left, constn 0] in
 let rhs = CN rec_pred [CN rec_power [constn 2, CN rec_lg [CN S [rec_right], constn 2]]] in
 let disj3 = CN rec_noteq [rec_right, rhs] in
 let disj4 = CN rec_eq [rec_right, constn 0] in
 CN rec_disj [CN rec_disj [CN rec_disj [disj1, disj2], disj3], disj4])"
+definition
+"rec_nostop = CN rec_nstd [rec_conf]"
+definition
+"rec_value = CN rec_pred [CN rec_lg [S, constn 2]]"
+definition
+"rec_halt = MN rec_nostop"
+definition
+"rec_uf = CN rec_value [CN rec_right [CN rec_conf [rec_halt, Id 2 0, Id 2 1]]]"
+section {* Correctness Proofs for Recursive Functions *}
+lemma entry_lemma [simp]:
+"rec_eval rec_entry [sr, i] = Entry sr i"
+by(simp add: rec_entry_def)
+lemma listsum2_lemma [simp]:
+"length xs = vl \<Longrightarrow> rec_eval (rec_listsum2 vl n) xs = Listsum2 xs n"
+by (induct n) (simp_all)
+lemma strt'_lemma [simp]:
+"length xs = vl \<Longrightarrow> rec_eval (rec_strt' vl n) xs = Strt' xs n"
+by (induct n) (simp_all add: Let_def)
+lemma map_suc:
+"map (\<lambda>x. Suc (xs ! x)) [0..<length xs] = map Suc xs"
+proof -
+have "Suc \<circ> (\<lambda>x. xs ! x) = (\<lambda>x. Suc (xs ! x))" by (simp add: comp_def)
+then have "map (\<lambda>x. Suc (xs ! x)) [0..<length xs] = map (Suc \<circ> (\<lambda>x. xs ! x)) [0..<length xs]" by simp
+also have "... = map Suc (map (\<lambda>x. xs ! x) [0..<length xs])" by simp
+also have "... = map Suc xs" by (simp add: map_nth)
+finally show "map (\<lambda>x. Suc (xs ! x)) [0..<length xs] = map Suc xs" .
+qed
+lemma strt_lemma [simp]:
+"length xs = vl \<Longrightarrow> rec_eval (rec_strt vl) xs = Strt xs"
+by (simp add: comp_def map_suc[symmetric])
+lemma scan_lemma [simp]:
+"rec_eval rec_scan [r] = r mod 2"
+by(simp add: rec_scan_def)
+lemma newleft_lemma [simp]:
+"rec_eval rec_newleft [p, r, a] = Newleft p r a"
+by (simp add: rec_newleft_def Let_def)
+lemma newright_lemma [simp]:
+"rec_eval rec_newright [p, r, a] = Newright p r a"
+by (simp add: rec_newright_def Let_def)
+lemma actn_lemma [simp]:
+"rec_eval rec_actn [m, q, r] = Actn m q r"
+by (simp add: rec_actn_def)
+lemma newstat_lemma [simp]:
+"rec_eval rec_newstat [m, q, r] = Newstat m q r"
+by (simp add: rec_newstat_def)
+lemma trpl_lemma [simp]:
+"rec_eval rec_trpl [p, q, r] = Trpl p q r"
+by (simp add: rec_trpl_def)
+lemma left_lemma [simp]:
+"rec_eval rec_left [c] = Left c"
+by(simp add: rec_left_def)
+lemma right_lemma [simp]:
+"rec_eval rec_right [c] = Right c"
+by(simp add: rec_right_def)
+lemma stat_lemma [simp]:
+"rec_eval rec_stat [c] = Stat c"
+by(simp add: rec_stat_def)
+lemma newconf_lemma [simp]:
+"rec_eval rec_newconf [m, c] = Newconf m c"
+by (simp add: rec_newconf_def Let_def)
+lemma conf_lemma [simp]:
+"rec_eval rec_conf [k, m, r] = Conf k m r"
+by(induct k) (simp_all add: rec_conf_def)
 lemma nstd_lemma [simp]:
 "rec_eval rec_nstd [c] = (if Nstd c then 1 else 0)"
 by(simp add: rec_nstd_def)
-text{*
-@{text "Nostop t m r"} means that afer @{text "t"} steps of
-execution, the TM coded by @{text "m"} is not at a stardard
-final configuration.
-*}
-fun Nostop :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"
-where
-"Nostop t m r = Nstd (Conf t m r)"
-definition
-"rec_nostop = CN rec_nstd [rec_conf]"
 lemma nostop_lemma [simp]:
 "rec_eval rec_nostop [t, m, r] = (if Nostop t m r then 1 else 0)"
 by (simp add: rec_nostop_def)
-fun Value where
-"Value x = (Lg (Suc x) 2) - 1"
-definition
-"rec_value = CN rec_pred [CN rec_lg [S, constn 2]]"
 lemma value_lemma [simp]:
 "rec_eval rec_value [x] = Value x"
 by (simp add: rec_value_def)
-text{*
+lemma halt_lemma [simp]:
-@{text "rec_halt"} is the recursive function calculating the steps a TM needs to execute before
+"rec_eval rec_halt [m, r] = Halt m r"
-to reach a stardard final configuration. This recursive function is the only one
+by (simp add: rec_halt_def)
-using @{text "Mn"} combinator. So it is the only non-primitive recursive function
-needs to be used in the construction of the universal function @{text "rec_uf"}.
+lemma uf_lemma [simp]:
-*}
+"rec_eval rec_uf [m, r] = UF m r"
+by (simp add: rec_uf_def)
-definition
-"rec_halt = MN rec_nostop"
-definition
-"rec_uf = CN rec_value [CN rec_right [CN rec_conf [rec_halt, Id 2 0, Id 2 1]]]"
-text {*
-The correctness of @{text "rec_uf"}, nonhalt case.
-*}
-subsection {* Coding function of TMs *}
-text {*
-The purpose of this section is to get the coding function of Turing Machine,
-which is going to be named @{text "code"}.
-*}
-fun bl2nat :: "cell list \<Rightarrow> nat \<Rightarrow> nat"
-where
-"bl2nat [] n = 0"
-| "bl2nat (Bk # bl) n = bl2nat bl (Suc n)"
-| "bl2nat (Oc # bl) n = 2 ^ n + bl2nat bl (Suc n)"
-fun bl2wc :: "cell list \<Rightarrow> nat"
-where
-"bl2wc xs = bl2nat xs 0"
-fun trpl_code :: "config \<Rightarrow> nat"
-where
-"trpl_code (st, l, r) = Trpl (bl2wc l) st (bl2wc r)"
-fun action_map :: "action \<Rightarrow> nat"
-where
-"action_map W0 = 0"
-| "action_map W1 = 1"
-| "action_map L = 2"
-| "action_map R = 3"
-| "action_map Nop = 4"
-fun action_map_iff :: "nat \<Rightarrow> action"
-where
-"action_map_iff (0::nat) = W0"
-| "action_map_iff (Suc 0) = W1"
-| "action_map_iff (Suc (Suc 0)) = L"
-| "action_map_iff (Suc (Suc (Suc 0))) = R"
-| "action_map_iff n = Nop"
-fun block_map :: "cell \<Rightarrow> nat"
-where
-"block_map Bk = 0"
-| "block_map Oc = 1"
-fun Goedel_code' :: "nat list \<Rightarrow> nat \<Rightarrow> nat"
-where
-"Goedel_code' [] n = 1"
-| "Goedel_code' (x # xs) n = (Pi n) ^ x * Goedel_code' xs (Suc n) "
-fun Goedel_code :: "nat list \<Rightarrow> nat"
-where
-"Goedel_code xs = 2 ^ (length xs) * (Goedel_code' xs 1)"
-fun modify_tprog :: "instr list \<Rightarrow> nat list"
-where
-"modify_tprog [] =  []"
-| "modify_tprog ((a, s) # nl) = action_map a # s # modify_tprog nl"
-text {* @{text "Code tp"} gives the Goedel coding of TM program @{text "tp"}. *}
-fun Code :: "instr list \<Rightarrow> nat"
-where
-"Code tp = Goedel_code (modify_tprog tp)"
 subsection {* Relating interperter functions to the execution of TMs *}
+lemma rec_step:
+assumes "(\<lambda> (s, l, r). s \<le> length tp div 2) c"
+shows "Trpl_code (step0 c tp) = Newconf (Code tp) (Trpl_code c)"
+apply(cases c)
+apply(simp only: Trpl_code.simps)
+apply(simp only: Let_def step.simps)
+apply(case_tac "fetch tp (a - 0) (read ca)")
+apply(simp only: prod.cases)
+apply(case_tac "update aa (b, ca)")
+apply(simp only: prod.cases)
+apply(simp only: Trpl_code.simps)
+apply(simp only: Newconf.simps)
+apply(simp only: Left.simps)
+oops
+lemma rec_steps:
+assumes "tm_wf0 tp"
+shows "Trpl_code (steps0 (1, Bk \<up> l, <lm>) tp stp) = Conf stp (Code tp) (bl2wc (<lm>))"
+apply(induct stp)
+apply(simp)
+apply(simp)
+oops
 lemma F_correct:
-assumes tp: "steps0 (1, Bk \<up> l, <lm>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"
+assumes tm: "steps0 (1, Bk \<up> l, <lm>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"
 and     wf:  "tm_wf0 tp" "0 < rs"
 shows "rec_eval rec_uf [Code tp, bl2wc (<lm>)] = (rs - Suc 0)"
+proof -
+from least_steps[OF tm]
+obtain stp_least where
+before: "\<forall>stp' < stp_least. \<not> is_final (steps0 (1, Bk \<up> l, <lm>) tp stp')" and
+after:  "\<forall>stp' \<ge> stp_least. is_final (steps0 (1, Bk \<up> l, <lm>) tp stp')" by blast
+have "Halt (Code tp) (bl2wc (<lm>)) = stp_least" sorry
+show ?thesis
+apply(simp only: uf_lemma)
+apply(simp only: UF.simps)
+apply(simp only: Halt.simps)
+oops
 end

changeset 250	745547bdc1c7
parent 249	6e7244ae43b8
child 256	04700724250f